Does the Langlands program preserve CFT's distinction between local and global theories? This question is vaguely related to: Different formulations of Class Field Theory
As I said there, I'm currently learning class field theory. For some motivation, I've also read a little about Langlands. Everything I've read about that program seems to be a generalization of the global formulation of CFT. Is the local language (and theorems) preserved as well? Are they obviously/with difficulty or/conjecturally equivalent?
 A: There is a local aspect to the Langlands program, known as the local Langlands correspondence.  In fact, Langlands conjectured the existence of such a correspondence for each local field $F$ and each reductive group $G$ over $F$.
He proved his local conjecture when $F$ is archimedean and $G$ is arbitrary.  The case when $G = \mathrm{GL}_n$ and $F$ is non-archimedean is now solved (by Laumon, Rapoport, and Stuhler in the function field case, and by Harris and Taylor in the case of $p$-adic fields).   There are many results known for other groups $G$ as well, but the full local conjectures for arbitrary $G$ are not yet settled, as far as I know.
In the $\mathrm{GL}_n$ case, the correspondence, roughly speaking, gives a bijeciton between (typically infinite dimensional!) irreducible representations of the group $\mathrm{GL}_n(F)$ and $n$-dimensional representations of the Galois group of $F$.  (So, unlike the abelian case, there is no isomorphism of groups, but rather a certain bijection between certain kinds of representations of rather different groups.)  
The case of general $G$ is more involved to state, and indeed, it is not so easy to find the precise conjecture in the literature.  In any case, it involves many complications, the most significant of which is probably so-called endoscopy.  Note that Ngo won the Fields medal this summer for his work on this topic.
Just as with local CFT, the motivations for the local conjecture come from the global theory.  On the one hand, representations of matrix groups over local fields arise naturally in the theory of automorphic forms (this is a generalization/reformulation of the classical theory of Hecke operators in the theory of modular forms), and, on the other hand, (at least certain) automorphic Hecke eigenforms are suppose to be related to representations of Galois groups of global fields by global reciprocity laws.  Restricting these global Galois representations to decomposition groups, one should recover the conjectural local correspondence.  (And as I noted in my answer to your earlier question, this is in fact the mechanism by which those local correspondences are normally constructed, in the cases when they can be constructed.)
Finally, I'm not sure if I understand your last question about equivalence correctly, but if you are asking whether the existence of the global Langlands correspondence (whatever exactly that means) should follow from the local correspondence, the answer is no.  Consider the case of CFT: knowing all the local Artin maps lets you write down the global Artin map, but you still have to prove the global Artin reciprocity law.  Similarly, knowing the local Langlands correspondence allows one to write down a candidate for the global correspondence, but to prove that this candidate actually does the job is another, even more difficult, matter.
(As one example: before the modularity theorem for elliptic curves was proved, people knew how to write down the 
candidate $q$-expansion that should be the weight $2$ modular form attached to an elliptic curve over $\mathbb Q$; this was because the relevant local issues were all completely understood.  The problem was then to prove that this actually was a weight $2$ modular form; this was a global issue, which was completely open until Wiles and Taylor, Breuil, Conrad and Diamond solved it.)
